Optimal. Leaf size=17 \[ -\frac {1}{6} \tanh ^{-1}\left (\frac {x}{2}\right )+\frac {1}{3} \tanh ^{-1}(x) \]
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Rubi [A]
time = 0.01, antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {1107, 213}
\begin {gather*} \frac {1}{3} \tanh ^{-1}(x)-\frac {1}{6} \tanh ^{-1}\left (\frac {x}{2}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 213
Rule 1107
Rubi steps
\begin {align*} \int \frac {1}{4-5 x^2+x^4} \, dx &=\frac {1}{3} \int \frac {1}{-4+x^2} \, dx-\frac {1}{3} \int \frac {1}{-1+x^2} \, dx\\ &=-\frac {1}{6} \tanh ^{-1}\left (\frac {x}{2}\right )+\frac {1}{3} \tanh ^{-1}(x)\\ \end {align*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(37\) vs. \(2(17)=34\).
time = 0.00, size = 37, normalized size = 2.18 \begin {gather*} -\frac {1}{6} \log (1-x)+\frac {1}{12} \log (2-x)+\frac {1}{6} \log (1+x)-\frac {1}{12} \log (2+x) \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(25\) vs.
\(2(11)=22\).
time = 0.02, size = 26, normalized size = 1.53
method | result | size |
default | \(-\frac {\ln \left (x +2\right )}{12}+\frac {\ln \left (x -2\right )}{12}-\frac {\ln \left (-1+x \right )}{6}+\frac {\ln \left (1+x \right )}{6}\) | \(26\) |
norman | \(-\frac {\ln \left (x +2\right )}{12}+\frac {\ln \left (x -2\right )}{12}-\frac {\ln \left (-1+x \right )}{6}+\frac {\ln \left (1+x \right )}{6}\) | \(26\) |
risch | \(-\frac {\ln \left (x +2\right )}{12}+\frac {\ln \left (x -2\right )}{12}-\frac {\ln \left (-1+x \right )}{6}+\frac {\ln \left (1+x \right )}{6}\) | \(26\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 25 vs.
\(2 (11) = 22\).
time = 0.27, size = 25, normalized size = 1.47 \begin {gather*} -\frac {1}{12} \, \log \left (x + 2\right ) + \frac {1}{6} \, \log \left (x + 1\right ) - \frac {1}{6} \, \log \left (x - 1\right ) + \frac {1}{12} \, \log \left (x - 2\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 25 vs.
\(2 (11) = 22\).
time = 0.36, size = 25, normalized size = 1.47 \begin {gather*} -\frac {1}{12} \, \log \left (x + 2\right ) + \frac {1}{6} \, \log \left (x + 1\right ) - \frac {1}{6} \, \log \left (x - 1\right ) + \frac {1}{12} \, \log \left (x - 2\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 26 vs.
\(2 (10) = 20\).
time = 0.07, size = 26, normalized size = 1.53 \begin {gather*} \frac {\log {\left (x - 2 \right )}}{12} - \frac {\log {\left (x - 1 \right )}}{6} + \frac {\log {\left (x + 1 \right )}}{6} - \frac {\log {\left (x + 2 \right )}}{12} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 29 vs.
\(2 (11) = 22\).
time = 4.83, size = 29, normalized size = 1.71 \begin {gather*} -\frac {1}{12} \, \log \left ({\left | x + 2 \right |}\right ) + \frac {1}{6} \, \log \left ({\left | x + 1 \right |}\right ) - \frac {1}{6} \, \log \left ({\left | x - 1 \right |}\right ) + \frac {1}{12} \, \log \left ({\left | x - 2 \right |}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.04, size = 11, normalized size = 0.65 \begin {gather*} \frac {\mathrm {atanh}\left (x\right )}{3}-\frac {\mathrm {atanh}\left (\frac {x}{2}\right )}{6} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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